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Theorem pjssmii 23166
Description: Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
pjidm.1  |-  H  e. 
CH
pjidm.2  |-  A  e. 
~H
pjsslem.1  |-  G  e. 
CH
Assertion
Ref Expression
pjssmii  |-  ( H 
C_  G  ->  (
( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) )  =  ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  A ) )

Proof of Theorem pjssmii
StepHypRef Expression
1 pjsslem.1 . . . . 5  |-  G  e. 
CH
2 pjidm.1 . . . . . 6  |-  H  e. 
CH
32choccli 22792 . . . . 5  |-  ( _|_ `  H )  e.  CH
41, 3chincli 22945 . . . 4  |-  ( G  i^i  ( _|_ `  H
) )  e.  CH
5 pjidm.2 . . . . 5  |-  A  e. 
~H
61, 5pjhclii 22907 . . . 4  |-  ( (
proj  h `  G ) `
 A )  e. 
~H
72, 5pjhclii 22907 . . . 4  |-  ( (
proj  h `  H ) `
 A )  e. 
~H
84, 6, 7pjsubii 23163 . . 3  |-  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
) )  =  ( ( ( proj  h `  ( G  i^i  ( _|_ `  H ) ) ) `  ( (
proj  h `  G ) `
 A ) )  -h  ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) )
94, 6pjhclii 22907 . . . . 5  |-  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  e.  ~H
104, 7pjhclii 22907 . . . . 5  |-  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) )  e.  ~H
119, 10hvsubvali 22506 . . . 4  |-  ( ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  -h  (
( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) )  =  ( ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  +h  ( -u 1  .h  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) ) )
12 inss1 3548 . . . . . . 7  |-  ( G  i^i  ( _|_ `  H
) )  C_  G
134, 5, 1pjss2i 23165 . . . . . . 7  |-  ( ( G  i^i  ( _|_ `  H ) )  C_  G  ->  ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  =  ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A ) )
1412, 13ax-mp 8 . . . . . 6  |-  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  =  ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )
152chshii 22713 . . . . . . . . . . . 12  |-  H  e.  SH
16 shococss 22779 . . . . . . . . . . . 12  |-  ( H  e.  SH  ->  H  C_  ( _|_ `  ( _|_ `  H ) ) )
1715, 16ax-mp 8 . . . . . . . . . . 11  |-  H  C_  ( _|_ `  ( _|_ `  H ) )
18 inss2 3549 . . . . . . . . . . . 12  |-  ( G  i^i  ( _|_ `  H
) )  C_  ( _|_ `  H )
194, 3chsscon3i 22946 . . . . . . . . . . . 12  |-  ( ( G  i^i  ( _|_ `  H ) )  C_  ( _|_ `  H )  <-> 
( _|_ `  ( _|_ `  H ) ) 
C_  ( _|_ `  ( G  i^i  ( _|_ `  H
) ) ) )
2018, 19mpbi 200 . . . . . . . . . . 11  |-  ( _|_ `  ( _|_ `  H
) )  C_  ( _|_ `  ( G  i^i  ( _|_ `  H ) ) )
2117, 20sstri 3344 . . . . . . . . . 10  |-  H  C_  ( _|_ `  ( G  i^i  ( _|_ `  H
) ) )
222, 5pjclii 22906 . . . . . . . . . 10  |-  ( (
proj  h `  H ) `
 A )  e.  H
2321, 22sselii 3332 . . . . . . . . 9  |-  ( (
proj  h `  H ) `
 A )  e.  ( _|_ `  ( G  i^i  ( _|_ `  H
) ) )
244, 7pjoc2i 22923 . . . . . . . . 9  |-  ( ( ( proj  h `  H
) `  A )  e.  ( _|_ `  ( G  i^i  ( _|_ `  H
) ) )  <->  ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) )  =  0h )
2523, 24mpbi 200 . . . . . . . 8  |-  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) )  =  0h
2625oveq2i 6078 . . . . . . 7  |-  ( -u
1  .h  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) )  =  ( -u 1  .h 
0h )
27 neg1cn 10051 . . . . . . . 8  |-  -u 1  e.  CC
28 hvmul0 22509 . . . . . . . 8  |-  ( -u
1  e.  CC  ->  (
-u 1  .h  0h )  =  0h )
2927, 28ax-mp 8 . . . . . . 7  |-  ( -u
1  .h  0h )  =  0h
3026, 29eqtri 2450 . . . . . 6  |-  ( -u
1  .h  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) )  =  0h
3114, 30oveq12i 6079 . . . . 5  |-  ( ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  +h  ( -u 1  .h  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) ) )  =  ( ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )  +h  0h )
324, 5pjhclii 22907 . . . . . 6  |-  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )  e.  ~H
33 ax-hvaddid 22490 . . . . . 6  |-  ( ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )  e.  ~H  ->  ( ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )  +h  0h )  =  ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A ) )
3432, 33ax-mp 8 . . . . 5  |-  ( ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )  +h  0h )  =  ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )
3531, 34eqtri 2450 . . . 4  |-  ( ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  +h  ( -u 1  .h  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) ) )  =  ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )
3611, 35eqtri 2450 . . 3  |-  ( ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  G ) `  A
) )  -h  (
( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( proj  h `  H ) `  A
) ) )  =  ( ( proj  h `  ( G  i^i  ( _|_ `  H ) ) ) `  A )
378, 36eqtri 2450 . 2  |-  ( (
proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
) )  =  ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  A )
381, 5pjclii 22906 . . . . 5  |-  ( (
proj  h `  G ) `
 A )  e.  G
39 ssel 3329 . . . . . 6  |-  ( H 
C_  G  ->  (
( ( proj  h `  H ) `  A
)  e.  H  -> 
( ( proj  h `  H ) `  A
)  e.  G ) )
4022, 39mpi 17 . . . . 5  |-  ( H 
C_  G  ->  (
( proj  h `  H
) `  A )  e.  G )
411chshii 22713 . . . . . 6  |-  G  e.  SH
42 shsubcl 22706 . . . . . 6  |-  ( ( G  e.  SH  /\  ( ( proj  h `  G ) `  A
)  e.  G  /\  ( ( proj  h `  H ) `  A
)  e.  G )  ->  ( ( (
proj  h `  G ) `
 A )  -h  ( ( proj  h `  H ) `  A
) )  e.  G
)
4341, 42mp3an1 1266 . . . . 5  |-  ( ( ( ( proj  h `  G ) `  A
)  e.  G  /\  ( ( proj  h `  H ) `  A
)  e.  G )  ->  ( ( (
proj  h `  G ) `
 A )  -h  ( ( proj  h `  H ) `  A
) )  e.  G
)
4438, 40, 43sylancr 645 . . . 4  |-  ( H 
C_  G  ->  (
( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) )  e.  G )
452, 5, 1pjsslem 23164 . . . . 5  |-  ( ( ( proj  h `  ( _|_ `  H ) ) `
 A )  -h  ( ( proj  h `  ( _|_ `  G ) ) `  A ) )  =  ( ( ( proj  h `  G
) `  A )  -h  ( ( proj  h `  H ) `  A
) )
462, 1chsscon3i 22946 . . . . . 6  |-  ( H 
C_  G  <->  ( _|_ `  G )  C_  ( _|_ `  H ) )
473, 5pjclii 22906 . . . . . . 7  |-  ( (
proj  h `  ( _|_ `  H ) ) `  A )  e.  ( _|_ `  H )
481choccli 22792 . . . . . . . . 9  |-  ( _|_ `  G )  e.  CH
4948, 5pjclii 22906 . . . . . . . 8  |-  ( (
proj  h `  ( _|_ `  G ) ) `  A )  e.  ( _|_ `  G )
50 ssel 3329 . . . . . . . 8  |-  ( ( _|_ `  G ) 
C_  ( _|_ `  H
)  ->  ( (
( proj  h `  ( _|_ `  G ) ) `
 A )  e.  ( _|_ `  G
)  ->  ( ( proj  h `  ( _|_ `  G ) ) `  A )  e.  ( _|_ `  H ) ) )
5149, 50mpi 17 . . . . . . 7  |-  ( ( _|_ `  G ) 
C_  ( _|_ `  H
)  ->  ( ( proj  h `  ( _|_ `  G ) ) `  A )  e.  ( _|_ `  H ) )
523chshii 22713 . . . . . . . 8  |-  ( _|_ `  H )  e.  SH
53 shsubcl 22706 . . . . . . . 8  |-  ( ( ( _|_ `  H
)  e.  SH  /\  ( ( proj  h `  ( _|_ `  H ) ) `  A )  e.  ( _|_ `  H
)  /\  ( ( proj  h `  ( _|_ `  G ) ) `  A )  e.  ( _|_ `  H ) )  ->  ( (
( proj  h `  ( _|_ `  H ) ) `
 A )  -h  ( ( proj  h `  ( _|_ `  G ) ) `  A ) )  e.  ( _|_ `  H ) )
5452, 53mp3an1 1266 . . . . . . 7  |-  ( ( ( ( proj  h `  ( _|_ `  H ) ) `  A )  e.  ( _|_ `  H
)  /\  ( ( proj  h `  ( _|_ `  G ) ) `  A )  e.  ( _|_ `  H ) )  ->  ( (
( proj  h `  ( _|_ `  H ) ) `
 A )  -h  ( ( proj  h `  ( _|_ `  G ) ) `  A ) )  e.  ( _|_ `  H ) )
5547, 51, 54sylancr 645 . . . . . 6  |-  ( ( _|_ `  G ) 
C_  ( _|_ `  H
)  ->  ( (
( proj  h `  ( _|_ `  H ) ) `
 A )  -h  ( ( proj  h `  ( _|_ `  G ) ) `  A ) )  e.  ( _|_ `  H ) )
5646, 55sylbi 188 . . . . 5  |-  ( H 
C_  G  ->  (
( ( proj  h `  ( _|_ `  H ) ) `  A )  -h  ( ( proj 
h `  ( _|_ `  G ) ) `  A ) )  e.  ( _|_ `  H
) )
5745, 56syl5eqelr 2515 . . . 4  |-  ( H 
C_  G  ->  (
( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) )  e.  ( _|_ `  H
) )
5844, 57jca 519 . . 3  |-  ( H 
C_  G  ->  (
( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
)  e.  G  /\  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
)  e.  ( _|_ `  H ) ) )
59 elin 3517 . . . 4  |-  ( ( ( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) )  e.  ( G  i^i  ( _|_ `  H ) )  <->  ( ( ( ( proj  h `  G
) `  A )  -h  ( ( proj  h `  H ) `  A
) )  e.  G  /\  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
)  e.  ( _|_ `  H ) ) )
606, 7hvsubcli 22507 . . . . 5  |-  ( ( ( proj  h `  G
) `  A )  -h  ( ( proj  h `  H ) `  A
) )  e.  ~H
614, 60pjchi 22917 . . . 4  |-  ( ( ( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) )  e.  ( G  i^i  ( _|_ `  H ) )  <->  ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
) )  =  ( ( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) ) )
6259, 61bitr3i 243 . . 3  |-  ( ( ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
)  e.  G  /\  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
)  e.  ( _|_ `  H ) )  <->  ( ( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
) )  =  ( ( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) ) )
6358, 62sylib 189 . 2  |-  ( H 
C_  G  ->  (
( proj  h `  ( G  i^i  ( _|_ `  H
) ) ) `  ( ( ( proj 
h `  G ) `  A )  -h  (
( proj  h `  H
) `  A )
) )  =  ( ( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) ) )
6437, 63syl5reqr 2477 1  |-  ( H 
C_  G  ->  (
( ( proj  h `  G ) `  A
)  -h  ( (
proj  h `  H ) `
 A ) )  =  ( ( proj 
h `  ( G  i^i  ( _|_ `  H
) ) ) `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3306    C_ wss 3307   ` cfv 5440  (class class class)co 6067   CCcc 8972   1c1 8975   -ucneg 9276   ~Hchil 22405    +h cva 22406    .h csm 22407   0hc0v 22410    -h cmv 22411   SHcsh 22414   CHcch 22415   _|_cort 22416   proj 
hcpjh 22423
This theorem is referenced by:  pjcji  23169  pjssmi  23651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cc 8299  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052  ax-addf 9053  ax-mulf 9054  ax-hilex 22485  ax-hfvadd 22486  ax-hvcom 22487  ax-hvass 22488  ax-hv0cl 22489  ax-hvaddid 22490  ax-hfvmul 22491  ax-hvmulid 22492  ax-hvmulass 22493  ax-hvdistr1 22494  ax-hvdistr2 22495  ax-hvmul0 22496  ax-hfi 22564  ax-his1 22567  ax-his2 22568  ax-his3 22569  ax-his4 22570  ax-hcompl 22687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-of 6291  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-2o 6711  df-oadd 6714  df-omul 6715  df-er 6891  df-map 7006  df-pm 7007  df-ixp 7050  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-fi 7402  df-sup 7432  df-oi 7463  df-card 7810  df-acn 7813  df-cda 8032  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-5 10045  df-6 10046  df-7 10047  df-8 10048  df-9 10049  df-10 10050  df-n0 10206  df-z 10267  df-dec 10367  df-uz 10473  df-q 10559  df-rp 10597  df-xneg 10694  df-xadd 10695  df-xmul 10696  df-ioo 10904  df-ico 10906  df-icc 10907  df-fz 11028  df-fzo 11119  df-fl 11185  df-seq 11307  df-exp 11366  df-hash 11602  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-rlim 12266  df-sum 12463  df-struct 13454  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-ress 13459  df-plusg 13525  df-mulr 13526  df-starv 13527  df-sca 13528  df-vsca 13529  df-tset 13531  df-ple 13532  df-ds 13534  df-unif 13535  df-hom 13536  df-cco 13537  df-rest 13633  df-topn 13634  df-topgen 13650  df-pt 13651  df-prds 13654  df-xrs 13709  df-0g 13710  df-gsum 13711  df-qtop 13716  df-imas 13717  df-xps 13719  df-mre 13794  df-mrc 13795  df-acs 13797  df-mnd 14673  df-submnd 14722  df-mulg 14798  df-cntz 15099  df-cmn 15397  df-psmet 16677  df-xmet 16678  df-met 16679  df-bl 16680  df-mopn 16681  df-fbas 16682  df-fg 16683  df-cnfld 16687  df-top 16946  df-bases 16948  df-topon 16949  df-topsp 16950  df-cld 17066  df-ntr 17067  df-cls 17068  df-nei 17145  df-cn 17274  df-cnp 17275  df-lm 17276  df-haus 17362  df-tx 17577  df-hmeo 17770  df-fil 17861  df-fm 17953  df-flim 17954  df-flf 17955  df-xms 18333  df-ms 18334  df-tms 18335  df-cfil 19191  df-cau 19192  df-cmet 19193  df-grpo 21762  df-gid 21763  df-ginv 21764  df-gdiv 21765  df-ablo 21853  df-subgo 21873  df-vc 22008  df-nv 22054  df-va 22057  df-ba 22058  df-sm 22059  df-0v 22060  df-vs 22061  df-nmcv 22062  df-ims 22063  df-dip 22180  df-ssp 22204  df-ph 22297  df-cbn 22348  df-hnorm 22454  df-hba 22455  df-hvsub 22457  df-hlim 22458  df-hcau 22459  df-sh 22692  df-ch 22707  df-oc 22737  df-ch0 22738  df-shs 22793  df-pjh 22880
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